JERUSALEM — A diplomatic push to end Israel’s nearly weeklong offensive in the Gaza Strip gained momentum Tuesday as Egypt’s president predicted that airstrikes would soon end, the U.S. Secretary of State raced to the region and Israel’s prime minister said his country would be a “willing partner” to a cease-fire with the Islamic militant group Hamas. As international diplomats worked to cement a deal, a senior Hamas official said an agreement was close even as relentless airstrikes and rocket attacks between the two sides continued. The Israeli death toll rose to five with the deaths Tuesday of an Israeli soldier and a civilian contractor. More than 130 Palestinians have been killed. “We haven’t struck the deal yet, but we are progressing and it will most likely be tonight,” Moussa Abu Marzouk said Tuesday from Cairo, where cease-fire talks were being held. A second Hamas official, Izzat Risheq, said later that a deal might not be reached. Israeli officials said only that “intensive efforts” were under way to end the fighting. Israeli media quoted Defense Minister Ehud Barak as telling a closed meeting that Israel wanted a 24-hour test period of no rocket fire to see if Hamas could enforce a truce.